On the island of Mumble, the Mumblian alphabet has only $5$ letters, and every word in the Mumblian language has no more than $3$ letters in it. How many words are possible? (A word can use a letter more than once, but $0$ letters does not count as a word.)
Solution: Often, the tricky part of casework problems is deciding what the cases should be. For this problem, it makes sense to use as our cases the number of letters in each word.

$\bullet$  Case 1: (1-letter words) There are $5$ 1-letter words (each of the $5$ letters is itself a 1-letter word).

$\bullet$  Case 2: (2-letter words) To form a 2-letter word, we have $5$ choices for our first letter, and $5$ choices for our second letter. Thus there are $5 \times 5 = 25$ 2-letter words possible.

$\bullet$  Case 3: (3-letter words) To form a 3-letter word, we have $5$ choices for our first letter, $5$ choices for our second letter, and $5$ choices for our third letter. Thus there are $5 \times 5 \times 5 = 125$ 3-letter words possible.

So to get the total number of words in the language, we add the number of words from each of our cases. (We need to make sure that the cases are exclusive, meaning they don't overlap. But that's clear in this solution, since, for example, a word can't be both a 2-letter word and a 3-letter word at the same time.)

Therefore there are $5 + 25 + 125 = \boxed{155}$ words possible on Mumble. (I guess the Mumblians don't have a lot to say.)